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    "# 单摆的物理公式\n",
    "\n",
    "## 1. 基本周期公式（小角度近似）\n",
    "\n",
    "### 周期公式\n",
    "$$ T = 2\\pi \\sqrt{\\frac{L}{g}} $$\n",
    "\n",
    "### 频率公式\n",
    "$$ f = \\frac{1}{T} = \\frac{1}{2\\pi} \\sqrt{\\frac{g}{L}} $$\n",
    "\n",
    "### 角频率公式\n",
    "$$ \\omega = \\frac{2\\pi}{T} = \\sqrt{\\frac{g}{L}} $$\n",
    "\n",
    "**其中：**\n",
    "- $ T $ = 周期（秒）\n",
    "- $ L $ = 摆长（米）\n",
    "- $ g $ = 重力加速度（约9.8 m/s²）\n",
    "- $ f $ = 频率（Hz）\n",
    "- $ \\omega $ = 角频率（rad/s）\n",
    "\n",
    "## 2. 位移方程\n",
    "\n",
    "### 角位移\n",
    "$$ \\theta(t) = \\theta_0 \\cos(\\omega t + \\phi) $$\n",
    "\n",
    "### 线位移\n",
    "$$ x(t) = L\\theta_0 \\cos(\\omega t + \\phi) $$\n",
    "\n",
    "**其中：**\n",
    "- $ \\theta_0 $ = 最大摆角（弧度）\n",
    "- $ \\phi $ = 初相位\n",
    "- $ t $ = 时间\n",
    "\n",
    "## 3. 速度公式\n",
    "\n",
    "### 角速度\n",
    "$$ \\omega_{\\theta} = \\frac{d\\theta}{dt} = -\\theta_0 \\omega \\sin(\\omega t + \\phi) $$\n",
    "\n",
    "### 线速度\n",
    "$$ v = L\\frac{d\\theta}{dt} = -L\\theta_0 \\omega \\sin(\\omega t + \\phi) $$\n",
    "\n",
    "## 4. 加速度公式\n",
    "\n",
    "### 角加速度\n",
    "$$ \\alpha = \\frac{d^2\\theta}{dt^2} = -\\theta_0 \\omega^2 \\cos(\\omega t + \\phi) $$\n",
    "\n",
    "### 线加速度\n",
    "$$ a = L\\alpha = -L\\theta_0 \\omega^2 \\cos(\\omega t + \\phi) $$\n",
    "\n",
    "## 5. 大角度摆动的精确解\n",
    "\n",
    "对于大角度摆动，周期需要用椭圆积分表示：\n",
    "\n",
    "$$ T = 4\\sqrt{\\frac{L}{g}} \\int_0^{\\frac{\\pi}{2}} \\frac{d\\phi}{\\sqrt{1 - k^2 \\sin^2 \\phi}} $$\n",
    "\n",
    "其中 $ k = \\sin(\\frac{\\theta_0}{2}) $\n",
    "\n",
    "## 6. 能量公式\n",
    "\n",
    "### 势能\n",
    "$$ U = mgL(1 - \\cos\\theta) $$\n",
    "\n",
    "### 动能\n",
    "$$ K = \\frac{1}{2}mL^2\\left(\\frac{d\\theta}{dt}\\right)^2 $$\n",
    "\n",
    "### 总机械能\n",
    "$$ E = mgL(1 - \\cos\\theta_0) $$\n",
    "\n",
    "## 7. 阻尼单摆\n",
    "\n",
    "有阻尼时的运动方程：\n",
    "$$ \\frac{d^2\\theta}{dt^2} + \\frac{b}{m}\\frac{d\\theta}{dt} + \\frac{g}{L}\\theta = 0 $$\n",
    "\n",
    "**其中：**\n",
    "- $ b $ = 阻尼系数\n",
    "- $ m $ = 摆球质量\n",
    "\n",
    "---\n",
    "\n",
    "## 📝 重要说明\n",
    "\n",
    "1. **小角度近似**：基本周期公式只在 $ \\theta < 10^\\circ $ 时准确\n",
    "2. **独立性**：周期与摆球质量无关（在理想条件下）\n",
    "3. **等时性**：小角度摆动时，周期与振幅无关\n",
    "\n",
    "> 这些公式是理解单摆运动的基础，在物理教学和工程应用中都有重要作用。"
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